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--- sav07_lecture_18 [2007/05/28 21:13]
vasu.singh
+++ sav07_lecture_18 [2007/05/29 00:37]
vasu.singh
@@ Line 27: / Line 27: @@
 This suggests that the two programs above are not equivalent for the purpose of reasoning
 in concurrently executing programs.
@@ Line 38: / Line 47: @@
      c ::= [r]  |  c[]c  |  c;c  |  c*  |  c||c  |  atomic(c)
-We express the semantics of c as the set of all possible sequences of relations of c.
+We express the semantics of c, $[[c]] \subseteq 2^{R^*}$ as the set of all possible sequences of relations of c.
 $[[c]] =  { r_1^1 r_2^1 \ldots r_{n1}^1, r_1^2 r_2^2 \ldots r_{n2}^2, \ldots, r_1^l r_2^2 \ldots r_{nk}^k}$.
-$[[ [r] ]] = \{ r \}$
+$[[\, [r]\, ]] = \{ r \}$
-$[[C1; C2]] = [[C1]] [[C2]] = \{r_1 \ldots r_n s_1 \ldots s_m | r_i \in [[C1]] and s_j \in [[C2]] \}$
+$[[C_1; C_2]] = [[C_1]] [[C_2]] = \{r_1 \ldots r_n s_1 \ldots s_m | r_i \in [[C_1]] and s_j \in [[C_2]] \}$
 (This is similar to a language theoretic approach, where we express the concatenation of two languages $L_1$ and $L_2$ as
 $L1.L2 = \{w_1 w_2 \mid w_1 \in L_1$ and $w_2 \in L_2 \}$.
+$[[C_1 [] C_2 ]] = [[C_1]] \cup [[C_2]]$.
+Let $A = \{bar, rab\}$ and $B = \{bara\}$. Then $A.B = \{barbara, rabbara\}$ and $A \cup B = \{bar, rab, bara\}$.
+$[[ C_1 || C_2 ]] = \bigcup {\{I(S_1,S_2)\}}$ where $S_1 = [[C_1]]$ and $S_2 = [[C_2]]$, where
+$I(S_1,S_2) = \{S_1^1, S_2^1, \ldots, S_1^n, S_2^n \mid S_1 = S_1^1 S_1^2 \ldots S_1^n$ and $S_2 = S_2^1 S_2^2 \ldots S_2^n \}$.
+$[[ C^* ]] = \cup_{n \ge 0} [[C^n]]$ where $[[C^0]] = \emptyset$ and $[[C^{n+1}]] = [[ C^n ; C]]$.
+$[[ $atomic$(C) ]] = $eval$([[C]])$ where eval is a function $eval: 2^{R^*} \rightarrow R$ such that
+eval$(\{r_1^1 \ldots r_{n1}^1, r_1^2 \ldots r_{n2}^2, \ldots, r_1^k \ldots r_{nk}^k \}) = (r_1^1 \circ r_1^2 \ldots \circ r_{n1}^1) \cup \ldots \cup (r_1^k \circ
+r_2^k \ldots \circ r_{nk}^k)$.
+An example
+  { b = x + y }
+  while (*) do
+    if (*) then
+      x++;
+      y--;
+    else
+      y++;
+      x--;
+    endif
+  end
+  ||
+  while (*) do
+    if (*) then
+      atomic{
+        x++;
+        y--;
+      }
+    else
+      atomic{
+        y++;
+        x--;
+      }
+    endif
+  end
+  { b = x + y }
 ==== Global reachability invariants ====